Almost invariant vectors and normalized positive type functions

Let $\xi$ be a unit vector and put $\phi (g)=⟨\pi (g)\xi ,\xi ⟩$. Unitarity and expansion of the squared norm give $\mathrm{\parallel }\pi (g)\xi -\xi {\mathrm{\parallel }}^2=2(1-\mathrm{Re}\phi (g))$, while [F3] gives $\mathrm{\mid }1-\phi (g)\mathrm{\mid }\le \mathrm{\parallel }\pi (g)\xi -\xi \mathrm{\parallel }$. By [F2], $\phi$ is a normalized continuous function of positive type.

If $\pi$ has almost invariant vectors, let $I=\{(Q,\epsilon ):Q\subseteq G\text{ compact},\epsilon >0\}$ and order it by $(Q,\epsilon )⪯(Q^{\mathrm{\prime }},{\epsilon }^{\mathrm{\prime }})$ when $Q\subseteq Q^{\mathrm{\prime }}$ and ${\epsilon }^{\mathrm{\prime }}\le \epsilon$. The index set is nonempty because $\mathrm{\varnothing }$ is compact. It is directed: for two indices, $Q\cup Q^{\mathrm{\prime }}$ is compact by [F5] and $\min (\epsilon ,{\epsilon }^{\mathrm{\prime }})>0$, so $(Q\cup Q^{\mathrm{\prime }},\min (\epsilon ,{\epsilon }^{\mathrm{\prime }}))$ is a common upper bound.

For every $i=(Q,\epsilon )\in I$, the witness set of unit vectors that are $(Q,\epsilon )$-invariant is nonempty by almost invariance. By AC [F8], choose one witness ${\xi }_i$ for each index. For any compact $K$ and $\eta >0$, set $i_0=(K,\eta \mathrm{/}2)$. If $i=(Q,\epsilon )⪰i_0$, then $K\subseteq Q$ and $\epsilon \le \eta \mathrm{/}2$, so for every $x\in K$, [F3] gives $\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }\le \mathrm{\parallel }\pi (x){\xi }_i-{\xi }_i\mathrm{\parallel }<\epsilon \le \eta \mathrm{/}2<\eta$. Thus the coefficient net converges to $1$ uniformly on compact subsets.

Conversely, suppose the coefficient net of unit vectors ${\xi }_i$ converges to $1$ uniformly on compact subsets. Fix compact $Q$ and $\epsilon >0$. Uniform convergence with tolerance ${\epsilon }^2\mathrm{/}2$ gives an index $i_0$ such that $\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }<{\epsilon }^2\mathrm{/}2$ for every $i⪰i_0$ and $x\in Q$. The identity in step 1.1 yields $\mathrm{\parallel }\pi (x){\xi }_i-{\xi }_i{\mathrm{\parallel }}^2=2(1-\mathrm{Re}{\phi }_i(x))\le 2\mathrm{\mid }1-{\phi }_i(x)\mathrm{\mid }<{\epsilon }^2$, so each such ${\xi }_i$ is $(Q,\epsilon )$-invariant. Taking ${\xi }_{i_0}$ supplies a witness for each given compact set and tolerance; hence $\pi$ has almost invariant vectors.

The estimate in step 2.1 used only eventual $(Q,\epsilon )$-invariance, not how the net was obtained. Therefore the coefficient net of any net of almost-invariant unit vectors also converges to $1$ uniformly on compact subsets.

For the GNS assertion, each normalized ${\phi }_i$ has ${\phi }_i(e)=1$, so [F6] gives a cyclic GNS triple with $\mathrm{\parallel }{\xi }_{{\phi }_i}\mathrm{\parallel }=1$ and coefficient ${\phi }_i$. Applying step 2.2's displacement estimate to the coefficient convergence shows that for each compact $Q$ and $\epsilon >0$, some $i_0$ has every ${\xi }_{{\phi }_i}$ $(Q,\epsilon )$-invariant for all $i⪰i_0$.

Embed ${\xi }_{{\phi }_{i_0}}$ as the vector supported in the $i_0$ coordinate of ${\widehat{⨁}}_{i\in I}{H}_{{\phi }_i}$. By [F7] this is a unit vector, and the direct sum action on that coordinate agrees with ${\pi }_{{\phi }_{i_0}}$. It is therefore $(Q,\epsilon )$-invariant. Since this works for every compact $Q$ and every $\epsilon >0$, the Hilbert direct sum has almost invariant vectors.

AC is used to select all witnesses in step 2.1 and is assumed by the GNS and direct-sum interfaces [F6]–[F8]. The estimates, the reverse implication in step 2.2, and the coordinate embedding use no further choice.